Boundedness in a attraction-repulsion chemotaxis system with nonlinear production

We study the quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type ut = ∆u−χ∇·(u∇υ)+ξ∇·(u∇ω) with nonlinear production 0 = ∆υ −βυ +αus, 0 = ∆ω −δω +γur, subject to the homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ ℝN (N ≥ 3) with smooth boundary. It is proved that for every α, β, δ, γ, χ, ξ > 0 and s ≥ N2,r > N−2/N s, there exists ξ∗ > 0 such that if ξ > ξ∗ and any sufficiently regular initial datum u0(x) ≥ 0, then the model has a unique global classical solution (u, υ, ω), which is bounded in Ω × (0, ∞).